1. The problem states that the displacement of the missile above the ground at time $t$ seconds is given by the function $$x(t) = -5t^2 + 100t + 10.$$ We need to estimate the average velocity of the missile from $t=0$ seconds to $t=6$ seconds. 2. The average velocity over an interval $[a,b]$ is given by the formula: $$\text{Average velocity} = \frac{x(b) - x(a)}{b - a}.$$ 3. Here, $a=0$ and $b=6$. We first calculate $x(0)$: $$x(0) = -5(0)^2 + 100(0) + 10 = 10.$$ 4. Next, calculate $x(6)$: $$x(6) = -5(6)^2 + 100(6) + 10 = -5(36) + 600 + 10 = -180 + 600 + 10 = 430.$$ 5. Now, compute the average velocity: $$\frac{x(6) - x(0)}{6 - 0} = \frac{430 - 10}{6} = \frac{420}{6} = 70.$$ 6. Therefore, the average velocity of the missile from $t=0$ to $t=6$ seconds is $70$ metres per second.